Nuprl Lemma : exp-minus
∀[n:ℕ]. ∀[x:ℤ]. ((-x)^n = if (n mod 2 =z 0) then x^n else -x^n fi ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
modulus: a mod n,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
true: True,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
nequal: a ≠ b ∈ T ,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
minus-one-mul,
exp_wf2,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
decidable__equal_int,
le_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
itermMinus_wf,
int_term_value_minus_lemma,
squash_wf,
true_wf,
exp-of-mul,
exp-minusone,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
isect_memberEquality,
axiomEquality,
minusEquality,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
setElimination,
rename,
dependent_set_memberEquality,
lambdaEquality,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
applyEquality,
imageElimination,
universeEquality,
multiplyEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbZ{}]. ((-x)\^{}n = if (n mod 2 =\msubz{} 0) then x\^{}n else -x\^{}n fi )
Date html generated:
2018_05_21-PM-01_05_21
Last ObjectModification:
2018_01_28-PM-02_01_53
Theory : num_thy_1
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