### Nuprl Lemma : rnexp-rless

`∀x,y:ℝ.  ((r0 ≤ x) `` (x < y) `` (∀n:ℕ+. (x^n < y^n)))`

Proof

Definitions occuring in Statement :  rleq: `x ≤ y` rless: `x < y` rnexp: `x^k1` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` all: `∀x:A. B[x]` implies: `P `` Q` natural_number: `\$n`
Definitions unfolded in proof :  req_int_terms: `t1 ≡ t2` rge: `x ≥ y` cand: `A c∧ B` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` subtract: `n - m` eq_int: `(i =z j)` true: `True` nequal: `a ≠ b ∈ T ` assert: `↑b` bnot: `¬bb` guard: `{T}` sq_type: `SQType(T)` bfalse: `ff` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` less_than': `less_than'(a;b)` le: `A ≤ B` so_apply: `x[s]` so_lambda: `λ2x.t[x]` and: `P ∧ Q` top: `Top` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` uimplies: `b supposing a` or: `P ∨ Q` decidable: `Dec(P)` squash: `↓T` sq_stable: `SqStable(P)` real: `ℝ` sq_exists: `∃x:A [B[x]]` rless: `x < y` nat: `ℕ` nat_plus: `ℕ+` prop: `ℙ` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  rless_transitivity2 rnexp-positive real_term_value_const_lemma real_term_value_var_lemma real_term_value_mul_lemma real_term_value_sub_lemma real_polynomial_null req-iff-rsub-is-0 itermMultiply_wf itermSubtract_wf rsub_wf rless-implies-rless rmul_preserves_rless rleq_weakening_equal rmul_functionality_wrt_rleq2 rless_functionality_wrt_implies rnexp-nonneg rleq_weakening_rless rnexp_unroll rless_functionality add-subtract-cancel subtract_wf int_term_value_add_lemma itermAdd_wf int_subtype_base int_formula_prop_eq_lemma intformeq_wf rmul_wf neg_assert_of_eq_int assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal equal_wf eqff_to_assert assert_of_eq_int eqtt_to_assert bool_wf eq_int_wf false_wf primrec-wf-nat-plus le_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf full-omega-unsat decidable__le sq_stable__less_than nat_plus_properties real_wf int-to-real_wf rleq_wf nat_plus_wf rless_wf nat_plus_subtype_nat rnexp_wf rlessw_wf
Rules used in proof :  productEquality inlFormation cumulativity instantiate promote_hyp equalitySymmetry equalityTransitivity productElimination equalityElimination independent_pairFormation voidEquality voidElimination isect_memberEquality intEquality int_eqEquality lambdaEquality dependent_pairFormation approximateComputation independent_isectElimination unionElimination imageElimination baseClosed imageMemberEquality independent_functionElimination addEquality setElimination rename natural_numberEquality because_Cache dependent_set_memberEquality sqequalRule hypothesis applyEquality hypothesisEquality isectElimination thin dependent_functionElimination sqequalHypSubstitution extract_by_obid cut introduction lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}x,y:\mBbbR{}.    ((r0  \mleq{}  x)  {}\mRightarrow{}  (x  <  y)  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}\msupplus{}.  (x\^{}n  <  y\^{}n)))

Date html generated: 2018_05_22-PM-01_33_01
Last ObjectModification: 2018_05_21-AM-00_08_06

Theory : reals

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