### Nuprl Lemma : add-remove-nth

`∀[T:Type]. ∀[L:T List]. ∀[n:ℕ||L||].  (let x,L' = remove-nth(n;L) in add-nth(n;x;L') ~ L)`

Proof

Definitions occuring in Statement :  add-nth: `add-nth(n;x;L)` remove-nth: `remove-nth(n;L)` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` spread: spread def natural_number: `\$n` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` 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` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` guard: `{T}` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` lelt: `i ≤ j < k` int_seg: `{i..j-}` remove-nth: `remove-nth(n;L)` add-nth: `add-nth(n;x;L)` firstn: `firstn(n;as)` nth_tl: `nth_tl(n;as)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` append: `as @ bs` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` select: `L[n]` subtract: `n - m` le_int: `i ≤z j` lt_int: `i <z j`
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 less_than_wf int_seg_wf length_wf list-cases product_subtype_list colength-cons-not-zero colength_wf_list le_wf subtract-1-ge-0 subtype_base_sq nat_wf set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf intformeq_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le list_wf nil_wf int_seg_properties length_of_nil_lemma length_of_cons_lemma list_ind_cons_lemma reduce_tl_cons_lemma lt_int_wf eqtt_to_assert assert_of_lt_int le_int_wf assert_of_le_int non_neg_length eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf list_ind_nil_lemma general_arith_equation1 lelt_wf false_wf add-is-int-iff decidable__lt select-cons-tl
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  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 sqequalRule independent_pairFormation Error :universeIsType,  axiomSqEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination promote_hyp hypothesis_subsumption productElimination Error :equalityIsType1,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate cumulativity intEquality equalityTransitivity equalitySymmetry imageElimination applyLambdaEquality Error :equalityIsType4,  addEquality applyEquality universeEquality voidEquality isect_memberEquality lambdaEquality dependent_pairFormation isect_memberFormation equalityElimination baseClosed closedConclusion baseApply pointwiseFunctionality dependent_set_memberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[n:\mBbbN{}||L||].    (let  x,L'  =  remove-nth(n;L)  in  add-nth(n;x;L')  \msim{}  L)

Date html generated: 2019_06_20-PM-01_33_04
Last ObjectModification: 2018_10_03-PM-11_01_00

Theory : list_1

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