### Nuprl Lemma : append-segment

`∀T:Type. ∀as:T List. ∀i:{0...||as||}. ∀j:{i...||as||}. ∀k:{j...||as||}.`
`  (((as[i..j-]) @ (as[j..k-])) = (as[i..k-]) ∈ (T List))`

Proof

Definitions occuring in Statement :  segment: `as[m..n-]` length: `||as||` append: `as @ bs` list: `T List` int_iseg: `{i...j}` all: `∀x:A. B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  segment: `as[m..n-]` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` int_iseg: `{i...j}` cons: `[a / b]` decidable: `Dec(P)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` firstn: `firstn(n;as)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` append: `as @ bs` nth_tl: `nth_tl(n;as)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` le: `A ≤ B` cand: `A c∧ B` subtract: `n - m` le_int: `i ≤z j` lt_int: `i <z j` true: `True`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 istype-less_than list-cases int_iseg_wf length_wf nil_wf product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf cons_wf istype-nat list_wf istype-universe nth_tl_nil list_ind_nil_lemma le_int_wf eqtt_to_assert assert_of_le_int reduce_tl_cons_lemma eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf length_of_cons_lemma non_neg_length int_iseg_properties add-is-int-iff false_wf first0 subtype_rel_list top_wf firstn_wf istype-false list_ind_cons_lemma lt_int_wf assert_of_lt_int less_than_wf squash_wf true_wf minus-one-mul add-commutes minus-add minus-minus add-associates add-swap zero-add
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :lambdaFormation_alt,  cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination voidEquality promote_hyp hypothesis_subsumption productElimination Error :equalityIstype,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase universeEquality equalityElimination cumulativity addEquality productEquality pointwiseFunctionality Error :productIsType,  multiplyEquality minusEquality imageMemberEquality

Latex:
\mforall{}T:Type.  \mforall{}as:T  List.  \mforall{}i:\{0...||as||\}.  \mforall{}j:\{i...||as||\}.  \mforall{}k:\{j...||as||\}.
(((as[i..j\msupminus{}])  @  (as[j..k\msupminus{}]))  =  (as[i..k\msupminus{}]))

Date html generated: 2019_06_20-PM-01_34_53
Last ObjectModification: 2019_01_02-PM-00_29_18

Theory : list_1

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