### Nuprl Lemma : rnexp_unroll

`∀[x:ℝ]. ∀[n:ℕ].  (x^n = if (n =z 0) then r1 if (n =z 1) then x else x^n - 1 * x fi )`

Proof

Definitions occuring in Statement :  rnexp: `x^k1` req: `x = y` rmul: `a * b` int-to-real: `r(n)` real: `ℝ` nat: `ℕ` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uall: `∀[x:A]. B[x]` subtract: `n - m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` 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` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` ge: `i ≥ j ` int_upper: `{i...}` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` rev_uimplies: `rev_uimplies(P;Q)` subtract: `n - m`
Lemmas referenced :  req_witness rnexp_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int int-to-real_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int int_upper_subtype_nat false_wf le_wf nat_properties nequal-le-implies zero-add int_upper_subtype_int_upper int_upper_properties rmul_wf subtract_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_wf nat_wf real_wf req_weakening rmul_comm req_functionality rnexp-req int_subtype_base rnexp_zero_lemma rmul-one
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename because_Cache natural_numberEquality lambdaFormation unionElimination equalityElimination sqequalRule productElimination independent_isectElimination equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination hypothesis_subsumption dependent_set_memberEquality independent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidEquality computeAll

Latex:
\mforall{}[x:\mBbbR{}].  \mforall{}[n:\mBbbN{}].    (x\^{}n  =  if  (n  =\msubz{}  0)  then  r1  if  (n  =\msubz{}  1)  then  x  else  x\^{}n  -  1  *  x  fi  )

Date html generated: 2017_10_03-AM-08_31_35
Last ObjectModification: 2017_07_28-AM-07_27_08

Theory : reals

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