Nuprl Lemma : rnexp-minus-one
∀n:ℕ. (r(-1)^n = if (n rem 2 =z 0) then r1 else r(-1) fi )
Proof
Definitions occuring in Statement :
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
all: ∀x:A. B[x],
remainder: n rem m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
int_nzero: ℤ-o,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_wf,
rnexp_wf,
int-to-real_wf,
exp_wf2,
eq_int_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
req-int,
exp-equal-one,
modulus-is-rem,
nequal_wf,
equal-wf-T-base,
exp-equal-minusone,
rem_bounds_1,
less_than_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
req_functionality,
rnexp-int,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
minusEquality,
natural_numberEquality,
remainderEquality,
because_Cache,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
baseClosed,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
dependent_pairFormation,
promote_hyp,
inrFormation,
independent_pairFormation,
dependent_set_memberEquality,
imageMemberEquality,
setElimination,
rename,
imageElimination,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}n:\mBbbN{}. (r(-1)\^{}n = if (n rem 2 =\msubz{} 0) then r1 else r(-1) fi )
Date html generated:
2017_10_03-AM-08_32_27
Last ObjectModification:
2017_07_28-AM-07_27_43
Theory : reals
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