Nuprl Lemma : exp-minusone
∀[n:ℕ]. ((-1)^n = if (n mod 2 =z 0) then 1 else -1 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,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
modulus: a mod n,
btrue: tt,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
exp: i^n,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
bnot: ¬bb
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
exp0_lemma,
equal_wf,
squash_wf,
true_wf,
ite_rw_true,
iff_weakening_equal,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
primrec-unroll,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
subtype_base_sq,
add-one-mod-2,
subtract-add-cancel,
mod_bounds,
modulus_wf,
nequal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
minusEquality,
imageMemberEquality,
baseClosed,
productElimination,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
impliesFunctionality,
instantiate,
cumulativity,
dependent_set_memberEquality,
addLevel,
applyLambdaEquality,
promote_hyp,
productEquality
Latex:
\mforall{}[n:\mBbbN{}]. ((-1)\^{}n = if (n mod 2 =\msubz{} 0) then 1 else -1 fi )
Date html generated:
2018_05_21-PM-01_05_15
Last ObjectModification:
2018_01_28-PM-02_01_49
Theory : num_thy_1
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