`∀[I:Interval]`
`  ∀[p:partition(I)]`
`    ∀i:ℕ||full-partition(I;p)|| - 1. r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p) `
`  supposing icompact(I)`

Proof

Definitions occuring in Statement :  partition-mesh: `partition-mesh(I;p)` full-partition: `full-partition(I;p)` partition: `partition(I)` icompact: `icompact(I)` interval: `Interval` rbetween: `x≤y≤z` rsub: `x - y` int-to-real: `r(n)` select: `L[n]` length: `||as||` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` rbetween: `x≤y≤z` and: `P ∧ Q` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` sq_stable: `SqStable(P)` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` subtype_rel: `A ⊆r B` partition: `partition(I)` full-partition: `full-partition(I;p)` sq_type: `SQType(T)` select: `L[n]` cons: `[a / b]` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` icompact: `icompact(I)` nat: `ℕ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` last: `last(L)` append: `as @ bs` list_ind: list_ind nil: `[]` right-endpoint: `right-endpoint(I)` pi2: `snd(t)` endpoints: `endpoints(I)` left-endpoint: `left-endpoint(I)` pi1: `fst(t)` ge: `i ≥ j `
Lemmas referenced :  adjacent-partition-points sq_stable__and rleq_wf int-to-real_wf rsub_wf select_wf real_wf full-partition_wf int_seg_properties subtract_wf length_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_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_add_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt subtract-is-int-iff intformless_wf itermSubtract_wf int_formula_prop_less_lemma int_term_value_subtract_lemma false_wf partition-mesh_wf sq_stable__rleq less_than'_wf nat_plus_wf squash_wf length_of_cons_lemma length-append length_of_nil_lemma add-subtract-cancel int_seg_wf partition_wf icompact_wf interval_wf decidable__equal_int subtype_base_sq int_subtype_base cons_wf right-endpoint_wf nil_wf lelt_wf left-endpoint_wf rbetween_wf true_wf select_append_front iff_weakening_equal append_wf le_wf length_append subtype_rel_list top_wf length-singleton intformeq_wf int_formula_prop_eq_lemma select_cons_tl general_arith_equation1 select-append lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot less_than_wf add-is-int-iff list-cases product_subtype_list non_neg_length
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation productElimination natural_numberEquality independent_isectElimination addEquality setElimination rename because_Cache dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp imageElimination baseApply closedConclusion baseClosed independent_functionElimination independent_pairEquality applyEquality minusEquality axiomEquality imageMemberEquality instantiate cumulativity dependent_set_memberEquality universeEquality equalityElimination hypothesis_subsumption

Latex:
\mforall{}[I:Interval]
\mforall{}[p:partition(I)]
\mforall{}i:\mBbbN{}||full-partition(I;p)||  -  1
r0\mleq{}full-partition(I;p)[i  +  1]  -  full-partition(I;p)[i]\mleq{}partition-mesh(I;p)
supposing  icompact(I)

Date html generated: 2017_10_03-AM-09_41_17
Last ObjectModification: 2017_07_28-AM-07_56_28

Theory : reals

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