### 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

Home Index