Nuprl Lemma : qexp-difference-factor
∀[a,b:ℚ]. ∀n:ℕ. ((a ↑ n - b ↑ n) = ((a - b) * Σ0 ≤ i < n. a ↑ i * b ↑ n - i + 1) ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qsum: Σa ≤ j < b. E[j],
qsub: r - s,
qmul: r * s,
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
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,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
so_apply: x[s],
true: True,
sq_type: SQType(T),
qsub: r - s,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
uiff: uiff(P;Q),
subtract: n - m,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
nat_wf,
rationals_wf,
qadd_wf,
qexp_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
qmul_wf,
int-subtype-rationals,
qsum_wf,
int_seg_subtype_nat,
false_wf,
subtract_wf,
int_seg_properties,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
int_seg_wf,
decidable__lt,
add-subtract-cancel,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
squash_wf,
true_wf,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_comm_qrng,
iff_weakening_equal,
prod_sum_r_q,
sum_unroll_hi_q,
sum_unroll_lo_q,
qexp-zero,
sum_shift_q,
less_than_wf,
not-lt-2,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
zero-add,
add_functionality_wrt_le,
le-add-cancel,
exp_unroll_q,
qmul_assoc_qrng,
qmul_ac_1_qrng,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
add-swap,
qmul_one_qrng,
mon_assoc_q,
qadd_comm_q,
qadd_ac_1_q,
qadd_inv_assoc_q,
qsub_wf,
exp_zero_q,
sum_unroll_base_q,
qmul_zero_qrng,
mon_ident_q,
subtract-add-cancel,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
isect_memberEquality,
isectElimination,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
minusEquality,
applyEquality,
productElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
functionEquality,
multiplyEquality
Latex:
\mforall{}[a,b:\mBbbQ{}]. \mforall{}n:\mBbbN{}. ((a \muparrow{} n - b \muparrow{} n) = ((a - b) * \mSigma{}0 \mleq{} i < n. a \muparrow{} i * b \muparrow{} n - i + 1))
Date html generated:
2018_05_22-AM-00_03_03
Last ObjectModification:
2017_07_26-PM-06_51_19
Theory : rationals
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