Nuprl Lemma : list-decomp-nat

`∀[T:Type]. ∀L:T List. ∀i:ℕ||L|| + 1.  ∃K,J:T List. ((L = (K @ J) ∈ (T List)) ∧ (||K|| = i ∈ ℤ))`

Proof

Definitions occuring in Statement :  length: `||as||` append: `as @ bs` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` add: `n + m` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` top: `Top` so_apply: `x[s]` implies: `P `` Q` int_seg: `{i..j-}` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cand: `A c∧ B` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  list_induction all_wf int_seg_wf length_wf exists_wf list_wf equal_wf append_wf length-append length_of_nil_lemma decidable__equal_int subtype_base_sq int_subtype_base int_seg_properties int_seg_subtype false_wf int_seg_cases satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf length_of_cons_lemma nil_wf cons_wf list_ind_nil_lemma equal-wf-base-T subtract_wf decidable__le intformnot_wf itermSubtract_wf intformeq_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_formula_prop_eq_lemma decidable__lt add-is-int-iff itermAdd_wf int_term_value_add_lemma lelt_wf list_ind_cons_lemma squash_wf true_wf iff_weakening_equal append_back_nil equal-wf-base equal-wf-T-base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality natural_numberEquality addEquality cumulativity hypothesis because_Cache productEquality applyLambdaEquality isect_memberEquality voidElimination voidEquality independent_functionElimination dependent_functionElimination setElimination rename unionElimination instantiate intEquality independent_isectElimination equalityTransitivity equalitySymmetry hypothesis_subsumption independent_pairFormation productElimination dependent_pairFormation int_eqEquality computeAll baseClosed dependent_set_memberEquality pointwiseFunctionality promote_hyp imageElimination baseApply closedConclusion applyEquality imageMemberEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}i:\mBbbN{}||L||  +  1.    \mexists{}K,J:T  List.  ((L  =  (K  @  J))  \mwedge{}  (||K||  =  i))

Date html generated: 2017_04_17-AM-08_45_02
Last ObjectModification: 2017_02_27-PM-05_04_53

Theory : list_1

Home Index