### Nuprl Lemma : rnexp-req

`∀[k:ℕ]. ∀[x:ℝ].  (x^k = if (k =z 0) then r1 else x * x^k - 1 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` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` sq_type: `SQType(T)` guard: `{T}` rnexp: `x^k1` eq_int: `(i =z j)` subtract: `n - m` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` 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` prop: `ℙ` subtype_rel: `A ⊆r B` real: `ℝ` reg-seq-nexp: `reg-seq-nexp(x;k)` has-value: `(a)↓` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat_plus: `ℕ+` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nequal: `a ≠ b ∈ T ` true: `True` fastexp: `i^n` efficient-exp-ext less_than: `a < b` squash: `↓T` bdd-diff: `bdd-diff(f;g)` int-to-real: `r(n)` reg-seq-mul: `reg-seq-mul(x;y)` int_nzero: `ℤ-o` absval: `|i|` respects-equality: `respects-equality(S;T)` sq_stable: `SqStable(P)` rev_uimplies: `rev_uimplies(P;Q)` cand: `A c∧ B`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis natural_numberEquality inhabitedIsType lambdaFormation_alt unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination because_Cache sqequalRule instantiate cumulativity intEquality dependent_functionElimination independent_functionElimination dependent_pairFormation_alt equalityIstype promote_hyp voidElimination hypothesis_subsumption independent_pairFormation dependent_set_memberEquality_alt approximateComputation lambdaEquality_alt int_eqEquality isect_memberEquality_alt universeIsType closedConclusion isectIsTypeImplies applyEquality callbyvalueReduce setEquality functionEquality multiplyEquality addEquality divideEquality baseClosed sqequalBase imageMemberEquality functionIsType imageElimination universeEquality minusEquality applyLambdaEquality setIsType pointwiseFunctionality baseApply remainderEquality functionIsTypeImplies lessCases axiomSqEquality

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

Date html generated: 2019_10_29-AM-09_34_33
Last ObjectModification: 2019_01_31-PM-09_59_48

Theory : reals

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