### Nuprl Lemma : firstn_decomp

`∀[T:Type]. ∀[j:ℕ]. ∀[l:T List].  (firstn(j - 1;l) @ [l[j - 1]] ~ firstn(j;l)) supposing (j - 1 < ||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]` 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 ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` le: `A ≤ B` true: `True` sq_type: `SQType(T)` 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` bool: `𝔹` unit: `Unit` it: `⋅` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf subtract_wf length_wf list_wf decidable__le false_wf not-ge-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel nat_wf decidable__equal_int subtype_base_sq int_subtype_base list_decomp decidable__lt not-lt-2 le_antisymmetry_iff list_ind_cons_lemma list_ind_nil_lemma first0 tl_wf subtype_rel_list top_wf not-equal-2 le-add-cancel2 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 le_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 independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom cumulativity equalityTransitivity equalitySymmetry imageElimination productElimination because_Cache unionElimination independent_pairFormation addEquality applyEquality voidEquality intEquality minusEquality universeEquality instantiate imageMemberEquality baseClosed baseApply closedConclusion equalityElimination dependent_pairFormation 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  -  1  <  ||l||  and  0  <  j)

Date html generated: 2017_04_14-AM-08_48_03
Last ObjectModification: 2017_02_27-PM-03_35_09

Theory : list_0

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