### Nuprl Lemma : select_concat_sum

`∀[T:Type]. ∀[ll:T List List]. ∀[i:ℕ||ll||]. ∀[j:ℕ||ll[i]||].  (ll[i][j] = concat(ll)[Σ(||ll[k]|| | k < i) + j] ∈ T)`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` select: `L[n]` length: `||as||` concat: `concat(ll)` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` 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` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` le: `A ≤ B` less_than': `less_than'(a;b)` nat: `ℕ` ge: `i ≥ j ` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` cand: `A c∧ B` int_iseg: `{i...j}` gt: `i > j`
Lemmas referenced :  int_seg_wf length_wf select_wf list_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_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_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma select_concat add-member-int_seg1 sum_wf int_seg_subtype_nat concat_wf lelt_wf subtract_wf false_wf sum_lower_bound le_wf non_neg_length itermMultiply_wf int_term_value_mul_lemma itermSubtract_wf int_term_value_subtract_lemma sum_split length_wf_nat itermAdd_wf int_term_value_add_lemma less_than_wf squash_wf true_wf length_concat iff_weakening_equal subtype_base_sq int_subtype_base sum1 equal_wf zero-add add-is-int-iff firstn_wf length_firstn subtype_rel_sets sum_functionality length_firstn_eq select_firstn decidable__or equal-wf-base or_wf decidable__equal_int intformor_wf intformeq_wf int_formula_prop_or_lemma int_formula_prop_eq_lemma sum_split+ subtract-is-int-iff zero-le-nat
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality cumulativity hypothesisEquality because_Cache hypothesis setElimination rename independent_isectElimination productElimination dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll imageElimination universeEquality isect_memberFormation axiomEquality applyEquality dependent_set_memberEquality lambdaFormation addEquality equalityTransitivity equalitySymmetry imageMemberEquality baseClosed independent_functionElimination instantiate pointwiseFunctionality promote_hyp baseApply closedConclusion hyp_replacement applyLambdaEquality productEquality setEquality

Latex:
\mforall{}[T:Type].  \mforall{}[ll:T  List  List].  \mforall{}[i:\mBbbN{}||ll||].  \mforall{}[j:\mBbbN{}||ll[i]||].
(ll[i][j]  =  concat(ll)[\mSigma{}(||ll[k]||  |  k  <  i)  +  j])

Date html generated: 2017_04_17-AM-08_50_50
Last ObjectModification: 2017_02_27-PM-05_12_23

Theory : list_1

Home Index