### Nuprl Lemma : cantor-interval-rleq

`∀[a,b:ℝ].  ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹].  ((fst(cantor-interval(a;b;f;n))) ≤ (snd(cantor-interval(a;b;f;n)))) supposing a ≤ b`

Proof

Definitions occuring in Statement :  cantor-interval: `cantor-interval(a;b;f;n)` rleq: `x ≤ y` real: `ℝ` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` real: `ℝ` cantor-interval: `cantor-interval(a;b;f;n)` pi1: `fst(t)` pi2: `snd(t)` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` int_seg: `{i..j-}` lelt: `i ≤ j < k` int_nzero: `ℤ-o` true: `True` sq_stable: `SqStable(P)` squash: `↓T` rneq: `x ≠ y` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf less_than'_wf rsub_wf cantor-interval_wf nat_plus_properties real_wf pi2_wf equal_wf pi1_wf_top nat_plus_wf int_seg_wf bool_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma le_wf nat_wf rleq_wf primrec0_lemma subtype_rel_dep_function int_seg_subtype false_wf subtype_rel_self eq_int_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int sq_stable__rleq decidable__lt lelt_wf int-rdiv_wf int_subtype_base true_wf nequal_wf radd_wf int-rmul_wf equal-wf-base rdiv_wf int-to-real_wf rless-int rless_wf rmul_preserves_rleq rmul_wf primrec-unroll rleq_functionality int-rdiv-req req_weakening uiff_transitivity rmul-rdiv-cancel2 rmul_comm radd_comm rmul_preserves_rleq2 rleq-int radd-preserves-rleq radd_functionality int-rmul-req req_transitivity req_inversion rmul-identity1 rmul-distrib2 rmul_functionality radd-int squash_wf radd_comm_eq iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination productElimination independent_pairEquality applyEquality because_Cache productEquality equalityTransitivity equalitySymmetry minusEquality axiomEquality functionEquality unionElimination dependent_set_memberEquality functionExtensionality equalityElimination promote_hyp instantiate cumulativity addLevel spreadEquality imageMemberEquality baseClosed imageElimination inrFormation addEquality universeEquality

Latex:
\mforall{}[a,b:\mBbbR{}].
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbB{}].    ((fst(cantor-interval(a;b;f;n)))  \mleq{}  (snd(cantor-interval(a;b;f;n))))
supposing  a  \mleq{}  b

Date html generated: 2017_10_03-AM-09_50_48
Last ObjectModification: 2017_07_28-AM-08_01_10

Theory : reals

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