### Nuprl Lemma : firstn_decomp2

`∀[T:Type]. ∀[j:ℕ]. ∀[l:T List].  (firstn(j - 1;l) @ [l[j - 1]] ~ firstn(j;l)) supposing ((j ≤ ||l||) and 0 < j)`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` select: `L[n]` length: `||as||` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` subtract: `n - m` natural_number: `\$n` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` subtract: `n - m` le: `A ≤ B` firstn: `firstn(n;as)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` lt_int: `i <z j` select: `L[n]` cons: `[a / b]` ifthenelse: `if b then t else f fi ` bfalse: `ff` btrue: `tt` append: `as @ bs` subtype_rel: `A ⊆r B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 le_wf length_wf list_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf decidable__equal_int subtype_base_sq int_subtype_base list_decomp decidable__lt intformeq_wf int_formula_prop_eq_lemma list_ind_cons_lemma list_ind_nil_lemma first0 tl_wf subtype_rel_list top_wf squash_wf true_wf length_tl iff_weakening_equal lt_int_wf bool_wf equal-wf-base assert_wf eqtt_to_assert assert_of_lt_int select-cons-tl eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot le_int_wf bnot_wf uiff_transitivity assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity equalityTransitivity equalitySymmetry imageElimination productElimination because_Cache unionElimination universeEquality instantiate applyEquality imageMemberEquality baseClosed baseApply closedConclusion equalityElimination promote_hyp

Latex:
\mforall{}[T:Type].  \mforall{}[j:\mBbbN{}].  \mforall{}[l:T  List].
(firstn(j  -  1;l)  @  [l[j  -  1]]  \msim{}  firstn(j;l))  supposing  ((j  \mleq{}  ||l||)  and  0  <  j)

Date html generated: 2017_04_17-AM-07_52_08
Last ObjectModification: 2017_02_27-PM-04_25_09

Theory : list_1

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