Nuprl Lemma : exp-equal-minusone
∀[x:ℤ]. ∀[n:ℕ]. uiff(x^n = (-1) ∈ ℤ;(x = (-1) ∈ ℤ) ∧ ((n mod 2) = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
exp: i^n,
modulus: a mod n,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
top: Top,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
le: A ≤ B,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
itermVar_wf,
intformnot_wf,
intformand_wf,
mod_bounds,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
not_wf,
bnot_wf,
assert_wf,
eq_int_wf,
exp-minusone,
exp-one,
assoced_elim,
neg_assoced,
iff_weakening_equal,
true_wf,
squash_wf,
assoced_wf,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
false_wf,
decidable__lt,
exp-assoced-one,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
itermConstant_wf,
intformeq_wf,
satisfiable-full-omega-tt,
nat_properties,
exp0_lemma,
subtype_base_sq,
decidable__equal_int,
nat_wf,
int-subtype-int_mod,
le_wf,
int_mod_wf,
subtype_rel_set,
less_than_wf,
modulus_wf_int_mod,
equal-wf-T-base,
int_subtype_base,
equal-wf-base,
exp_wf2,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
lemma_by_obid,
isectElimination,
intEquality,
hypothesisEquality,
minusEquality,
natural_numberEquality,
productEquality,
applyEquality,
baseClosed,
because_Cache,
dependent_set_memberEquality,
imageMemberEquality,
lambdaEquality,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setElimination,
rename,
unionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
voidEquality,
dependent_pairFormation,
computeAll,
lambdaFormation,
addEquality,
imageElimination,
universeEquality,
equalityEquality,
promote_hyp,
impliesFunctionality,
int_eqEquality,
equalityElimination
Latex:
\mforall{}[x:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. uiff(x\^{}n = (-1);(x = (-1)) \mwedge{} ((n mod 2) = 1))
Date html generated:
2016_05_15-PM-04_46_29
Last ObjectModification:
2016_01_16-AM-11_24_22
Theory : general
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